Can anyone guide me more about integration?

Question

Guide me about how to find the area of shaded region in integration

Answer 1

You didn't supply any figure or state which coordinate system you are using.

 

Assuming the Cartesian coordinates (x,y) here is how to do it.

 

Step 1:

Cut the region into parts called elementary regions

An elementary region of type I is bounded by lines x=a and x=b and two functions y=f(x) and y=g(x).

An elementary region of type II is bounded by lines y=a and y=b and two functions x=f(y) and x=g(y).

Functions f and g need to each be defined by a single expression in the interval [a,b]. That is, no piecewise functions.

Try to cut into as few elementary regions as possible.

 

Step 2:

Find the areas of these elementary regions.

Assuming that in the interval [a,b], f > g the areas are as follows.

A Type I region can be sliced with vertical slices and has area ∫_a^b [f(x)-g(x)] dx.

A Type II region can be sliced with horizontal slices and has area ∫_a^b [f(y)-g(y)] dy.

 

Step 3:

The answer is the sum of the areas in step two.

 

You may also encounter a problem where the boundary of the region asked is a loop given by an implicit equation in x and y. Then you must first find the parametric equations x(t) and y(t) for the loop. If the loop is traced once clockwise as t goes from 0 to T, then the area is

 

A = ∫ y dx = ∫₀^T y(t) x'(t) dt.